Hosoya Polynomials of the Capped Zig-Zag Nanotubes

Abstract

The Hosoya polynomial of a graph \(G\) is defined as \(H(G,x) = \sum\limits_{k\geq 0} d(G,k)x^k,\)
where \(d(G, k)\) is the number of vertex pairs at distance \(k\) in \(G\). The calculation of Hosoya polynomials of molecular graphs is a significant topic because some important molecular topological indices such as Wiener index, hyper-Wiener index, and Wiener vector, can be obtained from Hosoya polynomials. Hosoya polynomials of zig-zag open-ended nanotubes have been given by Xu and Zheng et al. A capped zig-zag nanotube \(T(p, q)[C, D; a]\) consists of a zig-zag open-ended nanotube \(T(p, q)\) and two caps \(C\) and \(D\) with the relative position \(a\) between \(C\) and \(D\). In this paper, we give a general formula for calculating the Hosoya polynomial of any capped zig-zag nanotube. By the formula, the Hosoya polynomial of any capped zig-zag nanotube can be deduced. Furthermore, it is also shown that any two non-isomorphic capped zig-zag nanotubes \(T(p, q)[C, D; a_1]\), \(T(p, q’)[C, D; a_2]\) with \(q’ \geq q^* \geq p+1\) have the same Hosoya polynomial, where \(q^*\) is an integer that depends on the structures of \(C\) and \(D\).